Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
DO NOT USE A CALCULATOR IN THIS QUESTION. Solve the equation (2+√5)x² = 4x+3(2-√5), giving your answers in the form a+b√5 where a and b are integers.
In this question all lengths are in centimetres and all angles are in radians. The diagram shows a circle with centre O and radius r. The points A ...
It is given that f(x) = 2ln(3x-4) for x > a.
The diagram shows part of the curve y = 3 + 4/(2x+1) and the straight line 3y = 2x+6. Find the area of the shaded region, giving your answer in exa...
Variables x and y are such that when lgy is plotted against √x a straight line passing through the points (1, 5) and (2.5, 8) is obtained. Show tha...
The function g is defined for 0° ≤ x ≤ 120° by g(x) = 2+4 cos 6x.
The diagram shows the graph of y = h(x) where h(x) = (x+a)²(b+cx) and a, b and c are integers. The curve meets the x-axis at the points (-2, 0) and...
The volume, V, of a sphere of radius r is given by V = (4/3)πr³. The volume of a sphere is increasing at a constant rate of 24 cm³ s⁻¹. Find the ra...
The diagram shows the graph of y = f(x) where f is defined by f(x) = 3x/√(5x+1) for 0 ≤ x ≤ 3.
In this question all lengths are in centimetres and all angles are in radians.
In the expansion of (ax + b/x²)⁹ where a and b are constants with a > 0, the term independent of x is -145152 and the coefficient of x⁶ is –6912. S...
The line with equation x+3y = k, where k is a positive constant, is a tangent to the curve with equation x²+y²+2y-9 = 0. Find the value of k and he...
A curve has equation y = 32x² + 1/(8x²) where x ≠ 0.
Variables x and y are related by the equation y = 2+tan(1-x) where 0 ≤ x ≤ π/2. Given that x is increasing at a constant rate of 0.04 radians per s...
Variables P and T are known to be connected by the relationship P = Abᵀ, where A and b are constants. Values of P are found for certain values of t...
A curve has equation y = cos(x/4) where x is in radians. The normal to the curve at the point where x = 4π/3 cuts the x-axis at the point P. Find t...
A particle travels in a straight line so that, t seconds after passing a fixed point, its velocity, v ms⁻¹, is given by v = e⁴ for 0 ≤ t ≤ 4, v = 1...
The diagram shows a triangle OAB. The point C is the mid-point of OA. The point D lies on CB such that CD : DB = 2 : 3. OC = c CB = b. The point E ...
1 (a) On the axes, sketch the graph of y=-=(x+2)(2x-1)(x+5), stating the intercepts with the axes. [Figure 1.1 - Graph axes with x and y labels, a...
2 DO NOT USE A CALCULATOR IN THIS QUESTION. The polynomial p is such that p(x) = 6x³ - 35x² +34x+45.
3 (a) Write 1+1g(x² - 1) - 21g(x-1), where x > 1, as a single logarithm to base 10. Give your answer in its simplest form. (b) Solve the equation 4...
4 (a) The first three terms, in ascending powers of x, in the expansion of (3+px)ⁿ are 243+810x+qx², where n, p and q are constants. Find the value...
5 (a) The diagram shows the graph of y = acos bx+c, for -360° ≤ x ≤ 360°, where a, b and c are constants. Find the values of a, b and c. [Figure 5...
6 Find ∫₄²(2/(2x-3) - 3/(3x-5)²) dx, giving your answer in exact form.
7 Given that 2+cotθ = 3x and sinθ = √y, find y in terms of x.
8 Solve the equation 4 sin²(2α-π/3) = 1 for -π/2 < α < π/2. Give your answers in terms of π.
9 (a) Solve the following simultaneous equations. ex+y × e3x-2y = 1 x²y = 256
10 In this question, all distances are in metres and time, t, is in seconds. A particle P is at a fixed point O at time t = 0. The velocity, v, of ...
11 The tangent to the curve y = (3x-1)¹/² at the point where x = 3 meets the coordinate axes at the points A and B. The point with coordinates (a, ...
12 (a) It is given that y = (ln 3x) / x² for x > 0. Find dy/dx. Give your answer in the form (A+B ln 3x) / x³, where A and B are integers. (b) Henc...
The diagram shows the graph of y = a sinbx+c for -360°≤x≤ 360°, where a, b and c are constants. Find the values of a, b and c. [Figure 1.1]
Given that log₃r+2log₉s = 8, find the value of rs.
Given that y = tan(x/2), find the exact value of dy/dx when x = π/3.
A team of 8 people is to be formed from 6 teachers, 5 doctors and 4 police officers.
DO NOT USE A CALCULATOR IN THIS QUESTION. In this question, all lengths are in centimetres. The diagram shows the trapezium ABCD. The lengths of AB...
In this question, all lengths are in metres and all angles are in radians. The diagram shows a circle with centre O and radius 5. The points A, B, ...
The line y = 3x-2 intersects the curve 2x²-xy+y² = 2 at the points A and B. The point C with coordinates (k, 7/8) lies on the perpendicular bisecto...
A curve has equation y = (3x²-5)^(1/2) / (x+4).
In this question, all distances are in metres and time, t, is in seconds.